I know the following definition of the $\delta$-function: Given $x\in\mathbb{R}$ we have \begin{equation}\label{eq1} \delta(x)=\frac{1}{2}\lim_{\epsilon\rightarrow 0}\epsilon|x|^{\epsilon-1}. \end{equation}
I came across a similar looking definition in higher dimensions. Given $x\in \mathbb{R}^4$ we supposedly have \begin{equation}\label{eq2} \delta^{(4)}(x)=\frac{1}{2\pi^2}\lim_{\epsilon\rightarrow 0} \epsilon |x|^{\epsilon-4}, \end{equation} where $|x|$ is the usual Euclidean norm on $\mathbb{R}^4$. I can imagine a generalisation to any dimension \begin{equation} \delta^{(D)}(x)=f(D)\lim_{\epsilon\rightarrow 0} \epsilon |x|^{\epsilon-D}. \end{equation} Is there a way to prove these $D$-dimensional identities from the one-dimensional one? I would have guessed that $f(D)$ is related to the area of the $D$-sphere $S_{D-1}=\frac{2\pi^{D/2}}{\Gamma(D/2)}$, but that doesn't quite match up for $D=1$.